Dr. Karen Uhlenbeck (Institute for Advanced Study, Princeton NJ)

08/30/2021  4:10pm

Abstract: 

This work is motivated by a preprint of Bill Thurston’s from 1994, which is still a preprint but still in circulation.  In it he outlines an approach to Teichmuller Space based on maps between hyperbolic surfaces, which minimize the maximum norm of directional derivatives of a map, or the Lipschitz constant.  As a warm-up to studying this theory from the analysis point of view, George Daskalopoulos and I studied the problem of finding functions from a hyperbolic surface M  into a circle S^1 which minimize the maximum of the derivative among all Lipschitz maps in a homotopy class.  These are extensively discussed in the analysis literature as infinity harmonic maps. The relevance of least gradient functions and transverse measures came as a surprise.  In this talk, I will try to explain some of the concepts, which were in fact very new to us, and outline the main results.  One of the main theorems is that the maximum stretch set (set on which the function takes on the maximum of the derivative) is a geodesic lamination.